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a question related to muti-variable normal distribution
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silicon2006@hotmail.com
science forum beginner


Joined: 19 May 2006
Posts: 6

PostPosted: Thu Jun 15, 2006 8:37 am    Post subject: a question related to muti-variable normal distribution Reply with quote

(1) If x1, x2, x3, ..., xn all follow the same normal distribution (a,
sigma),
does x=maximum(x1, x2, ..., xn) also follow normal distribution?

(2) If x=maximum(x1, x2, ..., xn) does follow normal distribution,
what is mean(x) and std(x)?
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john2
science forum beginner


Joined: 01 Feb 2006
Posts: 15

PostPosted: Thu Jun 15, 2006 1:08 pm    Post subject: Re: a question related to muti-variable normal distribution Reply with quote

silicon2006@hotmail.com wrote:
Quote:
(1) If x1, x2, x3, ..., xn all follow the same normal distribution (a,
sigma),
does x=maximum(x1, x2, ..., xn) also follow normal distribution?

(2) If x=maximum(x1, x2, ..., xn) does follow normal distribution,
what is mean(x) and std(x)?


A rank statistics problem [qv]. If x1..xn are independent, the CDF of
the largest sample is the n-th power of the CDF of a single sample and
the PDF is not normal, though it might approximate it.

john2
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Debashis Dash
science forum beginner


Joined: 16 Jun 2006
Posts: 1

PostPosted: Fri Jun 16, 2006 12:00 am    Post subject: Re: a question related to muti-variable normal distribution Reply with quote

It is true that the distribution is not normal (gaussian to be more
general), but you can easilly find its CDF and PDF:

(capital F is the CDF and f is the pdf and subscript denotes the variable)

Let Y=max(X_1, X_2, ..., X_N)

Then, F_Y(a)=Pr(Y<=a)

=Pr(max(X_1, X_2, ..., X_N)<=a)

=Pr(X_1<=a).Pr(X_2<=a)...Pr(X_N<=a)

={F_X(a)}^N

Hence, f_Y(a)=d/da(F_Y(a))=N(F_X(a))^(N-1).f_X(a)

Which is not a surprise.

DD

"john2" <john2@8889.fsnet.co.uk> wrote in message
news:e6rm4n$c7g$1@newsg4.svr.pol.co.uk...
Quote:
silicon2006@hotmail.com wrote:
(1) If x1, x2, x3, ..., xn all follow the same normal distribution (a,
sigma),
does x=maximum(x1, x2, ..., xn) also follow normal distribution?

(2) If x=maximum(x1, x2, ..., xn) does follow normal distribution,
what is mean(x) and std(x)?


A rank statistics problem [qv]. If x1..xn are independent, the CDF of the
largest sample is the n-th power of the CDF of a single sample and the PDF
is not normal, though it might approximate it.

john2
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Ray Koopman
science forum Guru Wannabe


Joined: 25 Mar 2005
Posts: 216

PostPosted: Fri Jun 16, 2006 7:48 am    Post subject: Re: a question related to muti-variable normal distribution Reply with quote

Debashis Dash wrote:
Quote:
,,,the distribution is not normal (gaussian to be more general)...

How do you see "gaussian" as more general than "normal"?
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Google

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